Asia Mathematika Volume: 5 Issue: 1 , (2021) Pages: 143 – 157 Available online at www.asiamath.org Existence of solutions for Caputo fractional q-differential equations Abdelkader Benali 1* , Houari Bouzid 2 , Mohamed Houas 3 , 1 Department of Mathematics Faculty of Exact and Computer Science, Hassiba Benbouali University Chlef 02000, Algeria, Laboratory of Mechanics and energy, NP, C00L03UN020120210001. Orchid iD: 0000-0001-6205-3499 2 Department of Mathematics Faculty of Exact and Computer Science, Hassiba Benbouali University Chlef 02000, Algeria, Laboratory of Mechanics and energy, NP, C00L03UN020120210001. 3 Department of Mathematics Khemis Miliana University, Laboratory FIMA, UDBKM, Algeria. Received: 27 Mar 2021 • Accepted: 20 Apr 2021 • Published Online: 30 Apr 201 Abstract: In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed. Key words: q-Caputo’s integral, Caputo’s fractional derivative, fractional differential equations, Existence, Fixed point theorem, Leray-Shauders alternative. 1. Introduction In this work, we discuss the existence and uniqueness of the solutions for multi-point boundary value problem of nonlinear fractional differential equations with two Caputo’s fractional orders    c D α q x (t)= f (t, x (t)) + AJ β q g (t, x (t)) , 0 <q< 1,t ∈ [0,T ] , x(0) = J 2-α q x (0) ,x (T )= BJ α-1 q x (T ) , (1) where D α is the fractional q-derivative of the Caputo type of orders α ∈]1, 2], J ϑ q is the Caputo’s fractional integral of order ϑ> 0,ϑ ∈{β, 2 - α, α - 1} , A, B are real constants and f, g : [0,T ] × R → R, are continuous functions on [0,T ].The existence results for the multi-point boundary value problem (1) are based on variety of fixed point theorems, such as Banach’s fixed point theorem, Leray-Schauder’s non-linear alternative and Leray-Schauder’s degree theory. In recent years, boundary value problems of fractional q- differential equations involving a variety of boundary conditions have been investigated by several researchers, see for example [4, 12, 17, 18, 20]. By using fixed point theory, many researchers have established existence results for some q- differential equations. For more details, we refer the reader to [17, 19, 23, 25, 26] and the references therein. Fractional differential equations have recently been studied by several researchers. For some earlier work on the topic, we refer to [6, 8–11, 15], whereas some recent work on the existence theory of fractional hybrid differential equations can be found in [13, 14, 16, 22, 24]. Recently, many researchers have studied the existence of solutions for some fractional ( q- fractional) difference equations. For some recent work on fractional differential equations, we refer to [4, 20, 23, 26] and the references therein. c Asia Mathematika, DOI: 10.5281/zenodo.4730073 * Correspondence: benali4848@gmail.com 143